Fourthly, the Maxwell equations in the Lagrangian coordinates appear stronger nonlinearities than that in the Eulerian coordinates.
This will bring us some trouble terms like \( \int _{\mathbb {R}} (\left| \phi _x \right| + \left| \psi _x \right|)(E_x^2 + b_x^2) \,\mathrm {d}x \)  in \( (\ref {Exbx4}) \)  when deriving the space integration term \(\Vert (\sqrt{\varepsilon }E_x,b_x)\Vert ^2\)  in Lemma REF .
To deal with these nonlinear terms we need the smallness of \( \Vert  \phi _x \Vert _{L^\infty } \) ,
which just requires the a priori assumption that \( \Vert  \phi \Vert _{H^2} \)  is small.
Comparing with the a priori assumption that \( \Vert  \phi \Vert _{H^1} \)  is small for the Navier-Stokes equations in [1]},
we still need to control the space integration term \( \Vert  \phi _{xx} \Vert ^2 \) .
This fact also urges us to improve the regularity of \( \psi \)  and \( \zeta \)  to the Sobolev space \( L^{\infty }\left(I ; H^{2}\right) \) ,
where \( I \)  is an interval with respect to the time variable \( t \) .
For this aim, we propose the a priori assumption that \( \Vert  \psi \Vert _{H^2} \)  and \( \Vert  \zeta \Vert _{H^2} \)  are both small.
